Lecture Notes in Microeconomics

Transcription

1 Lecture Notes in Microeconomics Lecturer: Adrien Vigier, University of Oslo Fall Foreword The aim of these notes is to provide a concise introduction to microeconomic modeling at the advanced undergraduate level No final year undergraduate student in economics is expected to find in these notes any concept or idea he is not already familiar with These old ideas will however be presented in a way that is likely to be new to most students at that stage in the course of their curriculum Some familiarity with the language of Mathematics will undoubtly help the reader of these notes While a number of key mathematical results are briefly presented in the appendix to these notes, these results are intended for immediate reference only In no way do they substitute an introductory course in real analysis Sydsaeter s Mathematical Analysis is an excellent reference for economists Those with a stronger background in Mathematics may want to use Rudin s Principles of Mathematical Analysis instead All introductory textbooks on microeconomics cover most of the material found in these notes, and indeed very often more than that Students are therefore encouraged to satisfy their curiosity by consulting alternative sources Rubinstein s outstanding Lecture Notes in Microeconomics for instance are freely available online Bear in mind however that any set of notes derives as much of its added value from what it chooses to leave out as from what it effectively contains Finally, while in principle usable on a stand-alone basis, these notes are primarily designed to support lectures Attending lectures should therefore help you improve your understanding of the material covered in the notes These notes are organized as follows Section 2 is devoted to the study of the consumer Section 21 elaborates a general framework in which to study issues related to consumption Section 22 illustrates some of the most important applications of the framework: intertemporal consumption (221), consumption under uncertainty (222), as well as labor supply 1

2 (223) We show in section 3 how our approach to consumption can be transferred over to think about production Section 4 introduces General Equilibrium Section 5 develops the concept of a financial asset That section unifies the applications covered in 22, and allows us to explicitly deal with temporal aspects of general equilibrium which constitute the theme of section 6 Results are divided into lemmas and propositions The lemmas tend to be purely technical results They are mere tools in the build-up to the propositions, in which the economic insights really lie Scattered in the text is also a series of questions While these questions are meant to give you an opportunity to exert the knowledge you have acquired, the results developed in them are often important complement to the material covered in the lectures As such they are part and parcel of these notes Sketch answers to all questions are provided in the Appendix More detailed answers will be given during the weekly seminars 2

3 2 The Consumer 21 General Framework Wherever possible, we will in these notes confine our analysis to a world containing two goods All results developed here naturally extend to higher dimensions, but our aim is to keep the analysis simple in order to focus on economic content Restricting attention to the 2-dimensional case also offers the great advantage of accommodating a complete graphical representation The ultimate foundation of our approach is the utility function, used to represent preferences 1 of the consumer over vectors of goods x = (x 1, x 2 ) A consumer having utility function u is one who prefers x to y iff u(x) > u(y) 2 Always bear in mind that utility functions are mere numerical tools used to represent underlying preferences In particular, if v is strictly increasing and a consumer has utility function u then v u is an equally valid utility function for that consumer A consumer s utility function is thus determined only up to an increasing transformation In principle, utility functions may take a variety of forms In order to make progress, one is bound to make certain restrictive assumptions regarding consumers preferences While these assumptions may to some extent be justified economically, their main asset is to greatly simplify the analysis of the model we build We will immediately state these assumptions in terms of utility functions You should convince yourself however that all these assumptions are preserved under any (smooth) increasing transformation This is critical, given we are claiming to make assumptions concerning consumers underlying preferences Our first assumption embodies the idea that consumers exhibit smooth preferences Assumption A1: u is C (u is smooth) Our second assumption embodies the idea that consumers always prefer consuming more Assumption A2: i u > 0, i (u is strictly increasing) 1 We could spend an entire course examining the relationship between consumers underlying preferences and the utility function representation of these preferences The interested reader is referred to Rubinstein s outstanding Lecture Notes in Microeconomics, freely available online 2 Notice that this immediately precludes certain preferences, in particular non-transitive preferences It is possible to show however that any well-behaved preferences can be represented using a utility function 3

4 Our third assumption embodies the idea that consumers prefer balanced baskets of goods 3 This is the natural assumption for complementary goods, but is also very compelling in the contexts of intertemporal consumption and consumption under uncertainty We explore these topics in detail later in the notes Assumption A3: u(λx + (1 λ)y) > min{u(x), u(y)}, λ (0, 1) (u is strictly quasiconcave) We will in these notes develop a set of results concerning utility functions satisfying the former assumptions Utility functions which fail to satisfy one or more of these assumptions have to be dealt with on an individual basis Definition 1 A consumer with utility function u is one who prefers x to y iff u(x) > u(y) A function u : R n R is a standard utility function iff it satisfies assumptions A1-A3 Henceforth, all utility functions we will be dealing with in these notes will be assumed to be standard utility functions, unless stated otherwise The following technical lemma records a useful way to check whether a given utility function satisfies the standard assumptions Lemma 1 Consider u such that A1-A2 hold Then assumption A3 holds iff u has convex level curves 4 Proof Suppose A3 holds, let L an arbitrary level curve of u (u(x) = l, x L), and f such that L = {(x, f(x)) : x R} We want to show that f is convex Let x and y belong to L By strict quasi-concavity u(λx + (1 λ)y) > min{u(x), u(y)} = l And, since u is strictly increasing, L lies below the line segment joining x and y But this implies f convex, since x and y were chosen arbitrarily For the converse, suppose u has convex level curves and let x and y be two consumption bundles Suppose moreover, wlog, that u(y) u(x) Let y the point on the same level curve as x such that y 2 = y 2 Note that y lies to the left of y, so that if we can show that u(λx + (1 λ)y ) > u(x) then we are done But the previous inequality indeed holds since the level curve containing x and y is decreasing, convex, and i u > 0, i, by A2 3 Observe that any concave function is also quasi-concave However, quasi-concavity is preserved under increasing transformation, while concavity is not So quasi-concavity is a statement about underlying preferences, while concavity is not 4 Recall that a level curve of u represents the set of points such that u(x) = l, for some fixed l 4

5 [Q1] Let α (0, 1) and consider the Cobb-Douglas functional form: u(x) = x α 1 x 1 α 2 Show that the Cobb-Douglas is (almost) a standard utility function We next introduce an important class of utility functions: Definition 2 A utility function u has the additive property iff there exists U : R R and (α i ) 1 i n, α i > 0, i, such that: u(x) = i α i U(x i ) (1) Utility functions satisfying the additive property are particularly convenient to handle and occupy for that reason a prominent place in microeconomic theory The following lemma shows for instance that assumptions A1-A3 take a particularly simple form for that class of utility functions Lemma 2 Let u satisfy the additive property, u(x) = i α iu(x i ) Then u satisfies A1-A3 iff 1 U is C 2 U > 0 3 U < 0 Proof Note first that 1 and 2 are evidently necessary and sufficient for A1 and A2 to hold, respectively We next show that if 3 holds then so does A3 We have u(λx + (1 λ)y) = i α i U(λx i + (1 λ)y i ) > i α ( λu(x i ) + (1 λ)u(y i ) ) (2) Hence u(λx + (1 λ)y) > λu(x) + (1 λ)u(y) We finally show that concavity of U is in fact a necessary condition for A3 to hold Suppose U not strictly concave We can thus find x such that U (x) 0 Let x 2 (x 1 ) denote the level curve of u passing through x = (x, x), so that α 1 U(x 1 ) + α 2 U(x 2 (x 1 )) = α 1 U(x) + α 2 U(x) (3) 5

6 Differentiating twice with respect to x 1 yields [ ] x α1 U (x 1 ) + α 2 U (x 2 )(x 2 = 2) 2 α 2 U (x 2 ) (4) In particular x 2(x) 0 By Lemma 1 therefore, u violates A3 [Q2] Using this time Lemma 2, show once more that the Cobb-Douglas is (almost) a standard utility function The problem of the consumer is straightforward to formulate on the basis of Definition 1 Let m denote the budget of a consumer with utility function u, and p the given vector of prices A consumer choosing his most preferred affordable bundle of goods effectively solves max u(x) st px m (5) x 0 Notice that if p i = 0 for some good i then, by A2, the consumer will demand an infinite amount of that good We will therefore suppose throughout that p i > 0, i Under this assumption, a solution to (5) always exists 5 There is moreover a unique optimal consumption bundle x Indeed, if there were two we could take a weighted average of them and, owing to the strict quasi-concavity of u, obtain a strictly higher utility level Note also that px = m at the optimum, by A2 We will say that the solution is interior if x i > 0, i Our first result embodies the following idea: prices establish the rate at which the consumer can trade goods against one another So unless this rate reflects his own valuation of the goods the consumer always has an incentive to trade goods at the market rate Thus, at an optimum, the consumer s marginal rate of substitution (MRS) must equal the price ratio The proposition also provides a useful set of sufficient conditions to elicit the optimal consumption bundle Proposition 1 Let p > 0 Any interior solution to (5) satisfies u/ x i u/ x j = p i p j (ie MRS = price ratio) (6) 5 All so-called existence results of microeconomics follow from standard results in topology These results are however beyond the scope of these notes 6

7 Moreover, if x satisfies (6) and px = m then x solves the consumer problem Proof Necessity of condition (6) is an immediate application of the Karush-Kuhn-Tucker Theorem (notice that we are here using the fact that the constraint is binding at an optimum) Suppose next x satisfies (6), and px = m By the Implicit Function Theorem the level curve of u at x is then tangent to the hyperplane px = m But utility functions have convex level curves by Lemma 1 The level curve of u at x thus lies entirely above its tangent This shows that x is optimal for the consumer problem We record for future reference the following definitions: Definition 3 The optimal consumption bundle in (5) is called Marshallian demand, and denoted x(p, m) Definition 4 The maximum utility level attained in (5) is called indirect utility, and denoted v(p, m) Definition 5 A good is said normal 6 iff the Marshallian demand for it satisfies x i (p, m)/ m 0 [Q3] Suppose two goods are perfect complements for a given consumer What functional form does this imply for this consumer s utility function? Comment What if the two goods are perfect substitutes instead? [Q4] Show that Marshallian demand is homogenous degree zero in (p, m), and satisfies px = m Show moreover that indirect utility is homogenous degree zero in (p, m), increasing in income, decreasing in prices, and quasiconvex in (p, m) [Q5] Let u a standard utility function and v : R R smooth and strictly increasing Apply (6) to v(u(x)) instead of u What do you observe? Comment [Q6] Let α (0, 1) and consider a consumer with Cobb-Douglas utility function Find the Marshallian demand functions and indirect utility function in this case Show that both goods are normal goods for this consumer 6 The definition is slightly misleading since a good may be normal for one consumer but not for another 7

8 [Q7] Let u(x) = x 1 +φ(x 2 ) Under what conditions is u a standard utility function? Assuming these conditions to be satisfied, find the Marshallian demand functions Comment Some careful inspection 7 will bring you to the observation that the solution to the consumer problem (5) is also the solution to a different closely related problem We next develop this intuition Consider the following problem and definitions: min px st u(x) u (7) x 0 Definition 6 The optimal consumption bundle in (7) is called Hicksian demand, and denoted h(p, u) Definition 7 The minimum expenditure attained in (7) is called expenditure function, and denoted e(p, u) [Q8] Show that the expenditure function is homogenous degree one in p, increasing in u as well as in prices, and concave in p The dual formulation of consumer theory rests on the following, mirror, results: Lemma 3 x(p, m) = h(p, v(p, m)) (8) Proof Let u = v(p, m) Note first that e ( p, u ) m Indeed, suppose we can find x such that u(x) u all the while px < m Since i u > 0, i, we can also find y close to x such that u(y) u and py < m But this contradicts u = v(p, m) So e ( p, u ) m Note also that e ( p, v(p, m) ) m since we have in particular px(p, m) = m, while u(x(p, m)) = u Hence e ( p, u ) = m The result follows since, once again, px(p, m) = m Lemma 4 h(p, u) = x(p, e(p, u)) (9) 7 Though admittedly mixed with a fair amount of intuition 8

9 Proof Let m = e(p, u) Note first that v(p, m) u Indeed, suppose we can find x such that u(x) > u and px m then we could also find y close to x such that u(y) > u and py < m But this implies e(p, u) < m, a contradiction So v(p, m) u But we also have v(p, m) u, since u ( h(p, u) ) = u while ph(p, u) = e(p, u) = m Hence v(p, m) = u The result now follows, since like we said already u ( h(p, u) ) = u and ph(p, u) = e(p, u) = m Corollary 1 m = e ( p, v(p, m) ) (10) Proof This was proven within the proof of lemma 3 Corollary 2 u = v ( p, e(p, u) ) (11) Proof This was proven within the proof of lemma 4 [Q9] Consider a consumer with Cobb-Douglas preferences Compute the Hicksian demand and expenditure functions in this case The dual formulation is somewhat more than a simple theoretical curiosity It is useful for at least two reasons It first allows us to illustrate income and substitution effects, and opens the way to comparative statics exercises Second, and maybe more importantly, it suggests an ingenious way of providing a unified treatment of consumption and production Leave production aside for now Our second result embodies the following idea: An increase in the price of good 1 makes good 1 relatively more expensive compared to other available goods In addition, it reduces the purchasing power of the consumer These effects substitution effect on the one hand, income effect on the other are essentially distinct effects The slutsky equation establishes this distinction formally and confirms that in so far as normal goods are concerned, the total effect on demand of an own-price increase is negative 9

10 Proposition 2 [Slutsky Equation] Suppose all goods are normal goods 8 For all i, j: x i (p, m) p j = h i(p, v(p, m)) x j (p, m) x i(p, m) p j m (12) In particular if i = j then both sides in (12) are negative Moreover, v(p, m)/ p i < 0 Proof By (9), and using the chain rule h i (p, u) p j = x i(p, e(p, u)) p j + x i(p, e(p, u)) e(p, u) (13) m p j By the Envelope Theorem, we also have e(p, u) p j = h j (p, u) (14) This establishes (12) Moreover, from (14): h i (p, u) p i = 2 e(p, u) p 2 i (15) But we have shown in [Q3] that e(p, u) is concave in p Hence h i(p,u) p i normal, both sides in (12) are negative < 0 and, if goods are That dv(p, m(p))/dp i < 0 is immediate since an increase in prices strictly reduces the set of goods affordable [Q10] Show that the substitution effect following an own-price increase is always negative The general framework we have developed remains silent on the source of the consumer s budget m As we will later see when using this framework for practical purposes, a recurrent scenario is one in which 9 m = m(p), ie in which wealth itself is a function of prices In the canonical example for instance, the consumer is originally endowed with a vector of goods e, 8 Observe that the Slutsky Equation holds for all goods, normal or not The sign however of the total effect of an own-price increase is determined only if the good is normal 9 Notice that m = m(p) allows for the possibility that m/ p = 0 So the truly general framework is the one in which m = m(p) I follow here however common practice and treat this case as an extension 10

11 so that he is both a seller and buyer of the goods he consumes Given prices p then: m = pe The consumer is a net buyer of good i iff x i > e i, and a net seller otherwise Naturally, extending the framework in this way creates important distortions regrading the effect of price changes and we must look beyond the Slutsky equation It is in particular no longer generally true that the total effect on demand of an own-price increase is negative If the consumer is a net seller of the good whose price increases he may benefit so much that he actually ends up raising his consumption of that good In a similar vein, the consumer need not be worse off following a price increase Indeed if he is a net seller of the good whose price increases then his old consumption bundle remains affordable, and he will even be left with a little cash in hand That consumer is thus in fact unambiguously better off Our next proposition formalizes these ideas Proposition 3 Suppose all goods normal and consider a consumer with endowment e, so that m = pe Then, for all i: 1 dx i(p,m(p)) dp i 2 dv(p,m(p)) dp i < 0 if x i > e i < 0 if x i > e i, dv(p,m(p)) dp i > 0 if e i > x i Proof Using the chain rule, we obtain the total derivative: dx i (p, m(p)) dp i = x i(p, m(p)) p i + x i(p, m(p)) m(p) (16) m p i We then obtain by (12) and the fact that m(p) p i = e i : [ dx i (p, m(p)) hi (p, v(p, m(p))) = x i (p, m(p)) x ] i(p, m(p)) dp i p i m + x i(p, m(p)) e i (17) m And, rearranging: dx i (p, m(p)) dp i = h i(p, v(p, m(p))) p i x ] i(p, m(p)) [x i (p, m(p)) e i m (18) This concludes the first part of the proof since, as already indicated in the proof of Proposition 2, h i(p,u) p i < 0 Suppose next e i > x i A rise in the price of good i leaves consumption bundle x strictly affordable Thus dv(p,m(p)) dp i > 0 in this case 11

12 Finally, suppose x i > e i Let u = v(p, m(p)) and assume for a contradiction that dv(p,m(p)) dp i 0, ie assume that the consumer remains able to reach level curve u following a rise in the price of good i By revealed preference the new consumption bundle must lie to the left of e But x i > e i, which implies that the demand for good i must be making a discontinuous jump This is impossible, owing to the continuity of demand (which itself is a consequence of the Maximum Theorem) 22 Applications 221 Intertemporal Consumption Consider a consumer with a two periods horizon, x i his consumption in period i, and U(x) his (ex post) utility from consuming quantity x 10 We are interested in the consumer s ex ante preferences over profiles of consumption over time x, represented by the (ex ante) utility function u While there are in principle infinitely many ways in which to define u, the time-discount assumption is by far the most convenient to work with 11 assumption sets u(x) = U(x 1 ) + δu(x 2 ), where δ 1 Let y i denote income received by the consumer in period i = 1, 2 The time-discount If the consumer is able to freely borrow and lend at the market interest rate r, his budget constraint is x 2 y 2 + (1 + r)(y 1 x 1 ): he can in period 2 consume his income from period 2 plus capital and 1 interest from any saving he has from period 1 Letting p = (1, ) we can then formulate the 1+r consumer problem as max u(x) st px py (19) x 0 Note that in case borrowing constraints apply we then have the additional condition that x 1 y 1 [Q11] Consider a consumer with a two periods horizon receiving income y i in period i = 1, 2, and such that u(x) = U(x 1 ) + δu(x 2 ) Show that consumption is greater in period 2 than in period 1 iff δ > (1 + r) 1 Can you say anything concerning the effect of changing the interest 10 We suppose here that this utility is time-independent 11 While extremely convenient analytically, bear in mind that the time-discount assumption is far from innocuous Indeed, it supposes important restrictions on consumers underlying preferences The fact for instance that cross-effects are excluded from the framework ( 2 u/ x 1 x 2 = 0) is hard to reconcile with certain aspects of human behavior such as habits 12

13 rate on saving? What if the utility function has the Cobb-Douglas functional form? [Q12] Consider a consumer with a two periods horizon receiving income y i in period i = 1, 2 Suppose we know that if the consumer can freely borrow and lend he consumes x 1 > y 1 Find that consumer s optimal consumption profile if now he can freely lend but cannot borrow money What if x 1 < y 1 instead? 222 Consumption under Uncertainty Uncertainty is conveniently illustrated by means of bets, or lotteries A bet B with n possible outcomes is a vector-pair (p, b) where p is the vector of probabilities and b the associated vector of contingent gains In particular µ(b), the expected gain of bet B, is then given by pb Definition 8 A bet is said to be fair iff it entails an expected gain/loss of 0, ie µ(b) = pb = 0 Let 1 denote the vector with all entries equal to 1 It is sometimes convenient to think of B as composed of a sure gain µ(b) on the one hand and a fair bet B 0 on the other hand, where b 0 = b µ(b)1 The attractiveness of this representation lies in the fact that we thereby separate the gain component of B from the risk component of B A risk-averse consumer for instance is one who would ideally like to part with the risk component of B Definition 9 A consumer is said to be risk-averse iff to any bet B = (p, b) he prefers the sure gain µ(b) The following definition is useful in the context of insurance, a theme we later investigate Definition 10 Let B = (p, b) The hedge of B is the bet B such that B = (p, b), B yielding b i whenever B yields b i [Q13] Consider a risk-averse consumer facing bet B, where µ(b) < 0 Show that B is worth strictly more than µ(b) for that consumer Comment We next introduce a very central concept in the study of consumption under uncertainty 13

14 Definition 11 Consider an uncertain environment with n contingencies and associated probabilities (p i ) 1 i n Let x i represent consumption under contingency i and u the utility function representing consumer preferences over vectors of contingent consumption A consumer is an expected-utility maximizer iff 12 there exists a function U such that u(x) = i p iu(x i ) The expected-utility assumption is quite compelling However, its greatest credential lies in the extent to which it simplifies the analysis of most problems 13 In particular, while riskaversion in no way invokes expected-utility, the latter phenomena takes a remarkably simple form under the expected-utility assumption Indeed, the following Lemma establishes that, in the expected-utility model, risk-averse consumers are precisely those with concave ex post utility function U Lemma 5 Consider an expected-utility maximizing consumer, and let U denote his ex post utility function The consumer is risk-averse iff U is concave Proof If U is concave then the consumer is risk-averse by Jensen s Inequality If U is not concave then we can find y, z, and λ such that U(λy + (1 λ)z) < λu(y) + (1 λ)u(z) But then consider bet B = (p, b) where p = (λ, 1 λ) and b = (y, z) We have u(b) = λu(y) + (1 λ)u(z) > U(λy + (1 λ)z) = u(i µ(b) ), where I µ(b) denotes the bet in which the consumer receives µ(b) for sure This shows that the consumer is not risk averse It is noteworthy to underline in lieu of concluding remark that Lemma 5, together with Lemma 2, show that in the context of expected-utility the general framework we developed in section 21 is, in fact, precisely a model of risk-aversion [Q14] Consider an expected-utility maximizing consumer Let U his ex post utility and m his wealth Show that the set of small bets he is willing to make expands according to U (m)/u (m) 12 In this definition p is treated as a parameter Of course, p may be a variable of the problem under study The interested reader is referred to Section While extremely convenient analytically, bear in mind that the expected-utility assumption is far from innocuous Indeed, it supposes important restrictions on consumers underlying preferences The Allais Paradox in particular indicates that human behavior is to some extent irreconcilable with the expected-utility assumption Section 81 elaborates on this important point 14

15 Application to the market for insurance: We saw in Q13 that risk-averse consumers are precisely those willing to pay a premium to hedge themselves against uncertain outcomes An insurance company is just another firm whose business model is to take advantage of this observation We develop here the simplest possible model of insurance Consider a consumer (or insuree), with initial wealth M, facing the risk of an accident occurring with probability q 14 If an accident occurs the consumer loses L M, otherwise nothing happens An insurance company offers to insure the consumer for a unit price p In other words, it offers for a price pk a contract specifying that the insurance company commits to pay the consumer amount K in case an accident occurs The object of what follows is to show that the problem of the consumer, namely choosing K, is in fact nothing more than a special case of the general problem we examined at length in section 21 Let x 1 denote consumption in case of no accident and x 2 denote consumption in case of accident We will start by showing that choosing K is equivalent to choosing x = (x 1, x 2 ) satisfying (1 p)x 1 + px 2 M pl (20) If the consumer purchases K units of insurance he is left with M pk if no accident takes place, ie he can choose x 1 M pk If an accident occurs on the other hand he will end up with M pk L + K = M L + (1 p)k He can thus choose to consume x 2 M L + (1 p)k Summing a weighted average of these inequalities yields (1 p)x 1 + px 2 (1 p)(m pk) + p(m L + (1 p)k) = M pl But note that this is exactly (20) Conversely, suppose that the consumer chooses (x 1, x 2) satisfying (20) and let us show that he can achieve this outcome with an appropriate choice of K Let K = M x 1, and check p that this choice of K works In case of no accident the consumer is left with M pk = M p M x 1 = x p 1, as desired In case of an accident he ends up with M L + (1 p)k = M L + (1 p) M x 1 He can thus consume x p 2 1M L 1 p p p x 1 But notice that the RHS of this inequality is at most x 2 since by assumption (x 1, x 2) satisfies (20) Since we have now shown that choosing K is equivalent to choosing x = (x 1, x 2 ) satisfying (20), this also means that the insurance problem can be stated as max x 0 u(x) such that (20) holds But notice that (20) can be written as px pe, where p = (1 p, p) and 14 We suppose here the accident probability given and, in particular, independent of the behavior of the insuree This circumvents the (important) topic of moral hazard 15

16 e = (M, M L) In other words, an equivalent formulation of the insurance problem is: max u(x) st px pe (21) x 0 The insurance problem can thus be framed as a standard consumer problem in which (i) the consumer s initial endowments are M in good times and M L in bad times and (ii) the consumer can trade consumption in good times for consumption in bad times at rate (1 p)/p 15 We conclude with the following useful definitions: Definition 12 An insurance policy is fair iff it entails a fair bet Definition 13 A consumer is said to be fully insured iff he bears no more risk, ie x i = x j for all i and j [Q15] Show that a risk-averse consumer always chooses full insurance whenever insurance is fair, ie whenever p = q Show moreover that the converse holds under the expected-utility assumption (ie the consumer does not choose full insurance if p q) [Q16] Consider an expected-utility maximizing risk-averse consumer with logarithmic ex-post utility function, and a risk-neutral insurance company Suppose the consumer s wealth is M but, with probability q he will lose L < M What is the set of pareto optimal contracts between consumer and insurer? Which of these are acceptable to both parties Which one of these entails a fair insurance policy? 223 Labour Supply [Q17] Suggest a way in which to approach labour supply using the general framework developed in section 21 Can you say anything concerning the effect of changing wages on labour supply? 15 ie give up p in good times and receive 1 p in bad times 16

17 3 The Producer We have in section 2 developed a theory of demand based on prices and consumers preferences This sections aims to outline the way in which a similar exercise can be carried out in view of developing a theory of supply based on prices and firms technological constraints One of the great qualities of the consumer model developed in these notes resides in the ease with which its formalism can be transferred over to production, thus yielding largely isomorphic approaches to consumption on the one hand and production on the other This section aims to elicit this isomorphism The starting point of a theory of production is the production function f such that to input vector z corresponds output y = f(z) It will once again prove useful to make certain assumptions concerning production functions We summarize these below: Assumption A1 : f is C Assumption A2 : i f > 0, i Assumption A3 : f is strictly concave The problem of profit maximization given output price p and input prices w is solves max py wz st f(z) y (22) (y,z) 0 Observe that a solution (y, z) to (22) is such that z minimizes the cost of producing y, ie min wz st f(z) y (23) z 0 But note that (23) is precisely the dual formulation of the consumer problem (7) in which x z, u() f(), p w, and u y Formally speaking the problem of the consumer and that of the producer are thus interchangeable Of course since in the case of the consumer it is the primal formulation which bears economic significance, while in the case of the producer it is the dual formulation, part of the analysis undertaken for the consumer looses relevance when carried over to the producer Consider Slutsky s equation for instance, eliciting changes in demand for a given level of expenditure Now, in the case of production, expenditure is a mere variable of the problem: Indeed firms care about profits, not costs So Slutsky s equation is of very little practical importance when discussing production 17

18 4 General Equilibrium I We developed in section 2 a theory of demand based on prices and consumers preferences, and in section 3 a theory of supply based on prices and firms technological constraints In both cases prices were taken as parameters of the problem under study In reality of course prices are endogenously determined, and the only parameters of the economy consist in consumers preferences and firms technological constraints General Equilibrium (GE) is the theory of prices It seeks to determine which prices will prevail in the economy and, as a consequence, which quantities of each goods will be produced and consumed It will be useful to think of an economy as a dynamical system A dynamical system (Ω, f) consists of a state space Ω and a process f describing the evolution of the system, such that if x t Ω denotes the state of the system in period t then x t+1 = f(x t ) (24) One of the most important concepts in the study of dynamical systems is that of an equilibrium An equilibrium of (Ω, f) is a state which, once reached, the system remains in indefinitely Observe from (24) that the set of equilibria thus coincides with the set of fixed points of the process f, ie the set of points such that f(x) = x It is very compelling to think of the economy as a dynamical system in which prices determine the state Somewhat peculiarly however, standard economic theory remains largely silent regarding the process describing the evolution of the system Instead, the theory characterizes equilibria at the outset, avoiding at that stage any reference to the underlying dynamic process It only later demonstrates as a consistency check, in a sense that the former equilibria in fact coincide with those of a well-defined dynamical process We will follow this practice here We will in these notes limit ourselves to the simplest form of GE models While supply in general results from production, we will take it to be fixed exogenously Such models are usually called pure exchange models A pure exchange model 16 begins with an endowment point E = ( A e, B e) At the outset, agent A (B) is endowed with goods-vector A e ( B e ) Such models have the great advantage of introducing supply and tying it up with consumers wealth in a very simple and convenient way Indeed, total supply is A e + B e, while the wealth of agent A (B) is given by p A e (p B e) 16 We limit ourselves here to two consumers in order to shorten and simplify the exposition 18

19 A market equilibrium is characterized by (i) prices p such that aggregate demand equals aggregate supply, and (ii) a consumption profile for each one of the consumers Formally: Definition 14 Let X = ( A x, B x) The pair (X, p) is a market equilibrium for the pure exchange economy with endowment E = ( A e, B e) if and only if A x + B x = A e + B e and each consumer s consumption profile maximizes his own utility function given prices p Let j x(p) denote the Marshallian demand of consumer j The vector j j x(p) j j e has an important economic interpretation It represents the aggregate excess demand vector given prices p We will use z(p) for a shorthand, so that z(p) = j x(p) j j j e (25) The following property is absolutely key It states that the value of the aggregate excess demand vector is always 0: z(p)p = 0 (26) [Q18] Prove (26) Observe, using (25), that a vector of prices p is an equilibrium price vector if and only if z(p) = 0 We will now show that the set of prices such that z(p) = 0 in fact corresponds to the equilibrium set of a well-defined dynamical system For technical reasons (that we will not dwell onto) we will, for the purposes of our next lemma, truncate the consumers Marshallian demands and set: j x i = j x i I { j x i j e i +ɛ} + ( j e i + ɛ)i { j x i > j e i +ɛ} (27) These truncated demands have the great advantage of being well-defined when prices are 0 It is moreover easy to show that a property corresponding to (26) holds, ie z(p)p = 0, for all p 17 Lastly, notice that p i = 0 implies z i > Each consumer must exhaust his budget constraint If he does not, this implies that we can find a good i such that p i > 0 and x i < e i But then, by (27), that consumer is not behaving optimally 18 If p i = 0 then by (27) each consumer demands strictly more of that good than he his endowed with There is thus strict excess demand for that good 19

20 Lemma 6 Let Σ denote the n-dimensional simplex, 19 and g : Σ Σ such that g i (p) = p i + max{0, z i (p)} j (p j + max{0, z j (p)}) (28) Then, g(p) = p if and only if z(p) = 0 Proof That z(p) = 0 implies g(p) = p is immediate We now show the converse Suppose, for a contradiction, that g(p) = p while z(p) 0 By (28), this implies that max{0, z i (p)} = kp i for some k 0, and all i In particular, one of the following must hold: 1 z i 0, i, with strict inequality for some i 2 z i > 0, i, and p i > 0, i 3 i such that z i 0 and p i = 0 We claim that none of the above can hold Suppose (1) holds Given zp = 0 this implies that there exists i such that z i < 0 while p i = 0 But this is impossible, by a former remark This also shows that (3) cannot hold Finally, (2) cannot hold since zp = 0 Lemma 6 is a central result It shows that a vector of prices p is a (market) equilibrium price vector if and only if it is an equilibrium state of the dynamical system (Σ, g) But given that Σ is a compact of R n, and g is continuous, we know that the dynamical system (Σ, g) has at least one equilibrium 20 It ensues that a market equilibrium does exist 21 [Q19] Let X = ( A x, B x), E = ( A e, B e), and p such that p i > 0 for all i Suppose A x ( B x) maximizes agent A s (B s) utility given prices p and wealth p A e (p B e) Suppose moreover that A x i + B x i = A e i + B e i in all but possibly one market Show that (X, p) is a market equilibrium for the pure exchange economy with endowment E [Q20] Consider a pure exchange economy, two consumers (A and B), two goods (1 and 2) Suppose consumer A has Cobb-Douglas utility function, while the two goods are perfect complements for consumer B Characterize market equilibrium if (i) A initially owns 10 units of 19 ie Σ = {x R n + i x i = 1} 20 This is a consequence of Brouwer s fixed point theorem 21 More precisely, lemma 6 shows that a market equilibrium exists given truncated demands a la (27) 20

21 good 1 and B 10 units of good 2 and (ii) A initially owns 10 units of good 2 and B 10 units of good 1 What is the set of pareto optimal allocations? Having formally defined market equilibria, we now aim to explore some of the properties these equilibria have The welfare theorems are the central results of GE Schematically, their message is the following Suppose S denotes the set of possible states of the economy, and P S the subset of states which are also pareto optimal Let s 0 S an arbitrary state, and s b the social planner s preferred state No matter how intricate the social planner s objective, the assumption that s b P is certainly compelling The First Welfare Theorem states that starting at s 0 and from there letting markets operate, the economy will eventually end in P 22 The Second Welfare Theorem states that if the social planner somehow manages to stir the economy to s b, he can then let markets operate and rest assured that the economy will remain in that state Somewhat ironically the welfare theorems are often perceived of as supporting arguments in favor of markets The First Welfare Theorem rests on the following observation: Prices fix the universal rate at which goods trade against one another and thereby also divide up the set of feasible allocations among consumers In equilibrium, each consumer chooses his preferred allocation within his subset Any improvement for one consumer must therefore lie in another consumer s subset and must, thereby, also be worse for that consumer owing to the fact that he chose to forego that allocation 23 The formal proof of the theorem requires some additional notation, which we now introduce Let F (E) denote the set of feasible consumption points 24 given endowment point E, ie F (E) = {X = ( A x, B x) : A x i + B x i = A e i + B e i, i} We call F (E) the Edgeworth box Let also A F (p, E) = {X F (E) : p A x p A e}, so that A F (p, E) is the subset of feasible consumption points affordable to consumer A under prices p Define B F (p, E) similarly Lemma 7 F (E) = A F (p, E) B F (p, E), for all p Proof Let X F (E) A F (p, E) Then 22 The Theorem remains silent however on how it will get from s 0 to P 23 By definition of a market equilibrium, each agent maximizes his own utility given prices There are thus no additional (mutual) gains from trade to be made at those prices The first welfare theorem notes that there are in fact no (mutual) gains from trade to be made at any prices 24 Strictly speaking, there are other feasible consumption points, namely those such that A x i + B x i < A e i + B e i for some i But these are not really interesting as they suppose that some resources go to waste In particular, it is cleat that none of these can be Pareto optimal 21

22 p B x = p( A e + B e A x) = p( A e A x) + p B e < p B e, ie X B F (p, E) Proposition 4 [First Welfare Theorem] A market equilibrium is pareto optimal Proof Let E the endowment point, p the equilibrium price vector, and X the equilibrium consumption point Suppose a feasible Y pareto dominates X, and suppose wlog that consumer A strictly prefers Y to X By definition of a market equilibrium Y / A F (p, E), and so by Lemma 7: Y B F (p, E) But this too is impossible, owing again to the definition of market equilibrium A last word of wisdom concerning the First Welfare Theorem Beware the following fallacious reasoning: in equilibrium each consumer maximizes his own utility so there are by definition no more gains from trade, and the First Welfare Theorem is a mere tautology The fault in this statement lies in the fact that there are by definition no more gains from trade at the equilibrium prices On the other hand what the First Welfare Theorem establishes is that there are in fact no more gains from trade at any prices [Q21] Argue that the reasoning used in the proof of the First Welfare Theorem collapses in the presence of externalities Proposition 5 [Second Welfare Theorem] Let X F (E) a pareto-optimal consumption point There exists p such that, in the pure exchange economy with endowment X, (X, p) is a market equilibrium Proof Let X = ( A x, B x) Let A L = {Y = ( A y, B y) F (E) : A u( A y) A u( A x)}, and B L = {Y = ( A y, B y) F (E) : B u( B y) B u( B x)} Note first that by quasi-concavity of utility functions both sets are convex We also have A L B L = {X} Indeed, if Y F (E) {X} belongs to A L B L, we can take a convex combination of X and Y and contradict the pareto-optimality of X So A L and B L are closed, convex sets with a singleton intersection The statement of the proposition is thus an immediate consequence of the Hyperplane Theorem 22

23 [Q22] Suppose (X, p) is a market equilibrium for the pure exchange economy with endowment E Can there exist an endowment point E and a price vector p p such that (X, p ) is a market equilibrium for the pure exchange economy with endowment E? What is the set of endowment points E such that (X, p) is a market equilibrium for the pure exchange economy with endowment E 5 Financial assets The analysis of general equilibrium developed in section 4 wholly disregards the issue of time In real life however, trade and consumption take place over time At any point in time therefore, individuals must decide whether to borrow or save all the while choosing what to consume at that point in time The key concept in this respect is that of a financial asset This section introduces financial assets, and sets the stage for section 6 in which we revisit GE with a view to incorporate temporal aspects We will throughout this section and the next consider T + 1 time periods, with t = 1 representing today and t > 1 representing future time periods We further suppose that for each t > 1 there are S possible contingencies, representing uncertainty about the future A pair (s, t) determines a state of the world Let W denote the set of all states of the world For expository purposes, it is useful to distinguish today s state of the world from future states We will let W 1 denote the set of all future states A financial asset is determined by a vector a R ST, specifying one payout for each possible future state of the world We suppose that the set of tradable assets is {a i } I i=1, so that I denotes the total number of tradable assets Let A denote the ST I matrix whose columns represent all tradable assets We refer to A as the asset structure We will henceforth make the assumption that I = ST 25 A portfolio ϕ R I is a combination of tradable assets, such that ϕ i indicates the quantity of asset i contained in the portfolio (note that we allow ϕ i < 0, i) The portfolio ϕ therefore induces payout vector Aϕ in future states We let q denote the price vector of the tradable assets at t = 1, so that, in particular, qϕ is the period 1 price of portfolio ϕ 25 The crucial assumption here is I ST If I < ST then rank(a) < ST, which most of the results derived in this section preclude The assumption that I actually equals ST is for simplification only 23

24 If the number of assets is large, different portfolios may induce identical payout vectors For a long time, one of financial traders main role was to seek out arbitrage opportunities between such portfolios Nowadays this is all computerized There are as a result, at any point in time, approximately no arbitrage opportunities remaining Formally: Definition 15 The no arbitrage condition states: Aϕ 1 = Aϕ 2 qϕ 1 = qϕ 2 (29) Intuitively, arbitrage creates a link between the prices of different assets We will now formalize this important idea Let ( e i) n i=1 denote the canonical basis in Rn If rank(a) = ST then for each i {1,, ST } we can find a portfolio v i such that Av i = e i (30) Definition 16 A family of portfolios ( v i) ST i=1 is a basis of elementary portfolios iff vi satisfies (30), for each i The following lemma provides a simple method which in practice allows us to find a basis of elementary portfolios Lemma 8 If rank(a) = ST and if moreover A is invertible then the column vectors of A 1 constitute a basis of elementary portfolios Proof Immediate from (30) Our next lemma summarizes the main practical consequence of the no arbitrage condition Lemma 9 Suppose the no arbitrage condition holds and rank(a) = ST Let ( v i) ST an i=1 arbitrary basis of elementary portfolios Then there exists π such that q i = πa i, i, or, in matrix form q = t Aπ (31) Moreover, π i = qv i, i (32) 24

25 Proof We have a i = j a i je j = j a i j(av j ) = A j a i jv j (33) On the other hand a i = Ae i (34) Hence, by no arbitrage q i = j a i jπ j = πa i (35) where π j = qv j (36) Naturally, any time the price of an elementary portfolio is lesser or equal than zero traders will demand an infinite amount of that portfolio, which in turn will push prices up We will thus henceforth assume π 0 [Q23] Let π 0 given by (31) Show that Aϕ 0 qϕ 0 Then show the converse, namely that Aϕ 0 qϕ 0 implies the existence of a vector π 0 such that q = t Aπ We next define an important class of portfolios Definition 17 Let τ {2,, T + 1} and b τ denote the portfolio paying out 1 if t = τ and 0 otherwise We call this portfolio a bond with maturity at date τ We define r τ such that 1 1+r τ indicates the price of this portfolio Notice that if ( v i) ST denotes a basis of elementary portfolios then: i=1 Ab τ = s Av sτ (37) By (31), we then have, using the no arbitrage condition: r τ = s π sτ (38) 25

26 In particular, we can define a probability distribution α τ such that, s {1,, S} π sτ = ατ s 1 + r τ (39) α τ s has a natural economic interpretation: we can view it as the probability which the market assigns to state of the world (s, τ) occurring 26 [Q24] Consider a world with two periods, today and tomorrow, and S states of the world Let A denote the asset structure of the economy, and q the asset price-vector Letting d = t (A 1 )q, show that r = 1 i d i i d i [Q25] A call option on a stock is a contract giving one the option to buy the stock at a specific price (the strike price) at a specified time in the future Consider a world with two future states (a good state and a bad state) and three financial assets There is first a bond, with interest rate r There is also a stock, whose price today is 1, and whose price tomorrow is u > 1 in the good state but d < 1 in the bad state There is finally a call option on the previous stock, with strike price K Assuming the no arbitrage condition holds, what is the price of the call option? (The formula you will obtain is a special case of the well-known Black-Scholes formula) What probability does the market assign to the good state occurring? In order to shed some light on financial assets role in the economy we will start with a simple model in which consumers only care about available wealth in each state of the world As usual, let u denote a utility function representing the consumer s preferences We suppose that the consumer receives income y w in each state of the world w W It will be useful to distinguish first-period income from later-periods income; we will write in this case y = (y 1, y 1 ) The consumer problem in this model is then: 27 max u(x) st ϕ,x 0 x1 y 1 qϕ ; x 1 y 1 + Aϕ (40) 26 To be more specific, suppose there exists a risk-neutral speculator who also happens to be fully unconstrained regarding his trading position If for some τ that speculator s beliefs do not coincide with α τ then he will take an infinite position 27 Notice that ϕ i in (40) may be positive or negative ϕ i < 0 is sometimes referred to as short-selling of asset i Finally, notice that the consumer is constrained in the speculative positions he may take Effectively, his income here plays the role of collateral 26

27 [Q26] Using the framework developed in the present section, re-formulate the intertemporal consumption problem as well as the insurance problem from sections 221 and 222, respectively In particular, specify A and q in each of these cases The purpose of the subsequent analysis is to show that (40) is in fact equivalent to (19), for an appropriate choice of p The idea is the following Consumers forego consumption today in order to buy assets which deliver future payouts; they then use these payouts to buy goods in the future Formally, we must therefore be able to by-pass financial assets Proposition 6 Suppose the no arbitrage condition holds and rank(a) = ST Let π 0 given by (31) and ξ = (1, π) Then x solves iff there exists ϕ such that (x, ϕ) solves (40) max u(x) st ξx ξy (41) x 0 Proof Note first that since (i) the objective functions are the same and (ii) all constraints are binding at any optimum of the two problems, it is enough to show that x Λ ϕ st (x, ϕ) Γ, where Γ = {(x, ϕ) : x 1 = y 1 qϕ ; x 1 = y 1 + Aϕ} ; Λ = {x : ξx = ξy} (42) We will first show that (x, ϕ) Γ x Λ We will then show that x Λ,! ϕ st (x, ϕ) Γ Let thus (x, ϕ) Γ Using (31) to substitute yields x 1 = y 1 πaϕ (43) We then have And, by definition of ξ x 1 = y 1 π ( x 1 y 1) (44) ξx = ξy (45) Conversely, let x Λ Since rank(a) = ST we can find a unique ϕ such that x 1 = 27